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arXiv · 2610.40084

Multi-excited random walks on general trees

Jack Mills·Tuan-Minh Nguyen
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Abstract

We study multi-excited random walks on infinite locally finite trees. The model is described by placing a stack of ``cookies'' at each vertex, with one cookie consumed on each successive visit. As long as cookies remain, the walk is in an excited regime and makes nearest-neighbour steps according to a prescribed sequence of biases. Once the stack is exhausted, the walk is in a non-excited regime and moves uniformly among the neighbouring vertices. The model is closely connected to foraging ecology, where repeated visits to a location can deplete local resources and thereby influence the attractiveness of that location for future visits. We prove that the process exhibits a sharp phase transition between recurrence and transience. The critical threshold is determined by the branching-ruin number of the tree, which is equal to the Hausdorff dimension of the tree's boundary under a suitable metric. Our result substantially extends Zerner's recurrence-transience criterion for multi-excited random walks on $\mathbb Z$ [Probab. Theory Related Fields 133 (2005)] to general trees.

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Jack Mills, Tuan-Minh Nguyen. 2026-10-29. Multi-excited random walks on general trees. https://arxiv.org/abs/2610.40084

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